In fact, it suffices that \(\sqrt{s}\,\mathbf {S}\)  has independent mean zero unit variance sub-Gaussian entries. From a standard result on the concentration of sub-Gaussian matrices [1]}, there exists a constant \(c\ge 1\)  such that, with probability \(1 - 2\exp (-t^2)\) ,
\(\Vert ()^{-1/2}{\hat{}}() ()^{-1/2}-\Vert \le \delta \vee \delta ^2 \quad \quad \text{for any }\quad \delta \ge c{\sqrt{d/s} + t/\sqrt{s}}.\) 
